There are thirteen 2-digit consecutive odd numbers. If 39 is the mean of the first five such numbers, then what is the mean of all the thirteen numbers?

Correct Answer: 47

Step-by-Step Explanation:

Step 1: Understand consecutive odd numbers and their mean

Consecutive odd numbers form an arithmetic progression where the common difference between consecutive terms is always 2.

For any odd number of terms in an arithmetic progression, the mean (average) of the terms is equal to the middle term.

Step 2: Determine the first term using the mean of the first five numbers

We are given that the mean of the first 5 consecutive odd numbers is 39.

Since there are 5 terms (an odd number of terms), the middle term—which is the 3^rd term—is equal to the mean.

Therefore, the 3^rd number is 39.

Let the first 5 consecutive odd numbers be:

$a,a+2,a+4,a+6,a+8$

The 3^rd term is $a+4=39$.

Solving for the first term $a$:

$a=39-4=35$

Step 3: Find the mean of all thirteen numbers

There are 13 consecutive odd numbers in total.

Since 13 is an odd number, the mean of all 13 consecutive terms is simply the middle term, which is the 7^th term.

The formula for the 7^th term of the sequence is:

$\text{7th term}=a+(7-1)\times 2=a+12$

Substituting $a=35$ into the equation:

$\text{Mean}=35+12=47$

Thus, the mean of all thirteen consecutive odd numbers is 47.